In those days a decree went out from Emperor Augustus that all the world should be registered. This was the first registration and was taken while Quirinius was governor of Syria. All went to their own towns to be registered. Joseph also went from the town of Nazareth in Galilee to Judea, to the city of David called Bethlehem, because he was descended from the house and family of David. He went to be registered with Mary, to whom he was engaged and who was expecting a child. While they were there, the time came for her to deliver her child. And she gave birth to her first born son and wrapped him in bands of cloth, and laid him in a manger, because there was no place for them in the inn.
Luke is our drama maven. You can tell because he begins the chapter with a big honking exaggeration. "All the world" should be registered? Really, now, the Roman Empire was big, but let's not be ridiculous here. On the other hand, if you were an average joe living in the Empire, I suppose it might have seemed the whole world to you.
Considering the buildup of chapter 1, this might seem to be a bit of an understated narrative of the birth, not unlike Matthew's. Of course Luke has more up his sleeve. Eventually Luke will give us the only surviving canonical account of Jesus as a child -- not infant -- by the end of this chapter.
For now, though, block out the other stuff, and let these eight verses sink in. The registration is called, and everybody has to go to the family home town to get counted properly. Joseph, briefly mentioned in Luke 1:27 (just before Mary gets to have a non-dream conversation with an angel and go hang with her old pregnant cousin Elizabeth), leads the family-to-be on the road to Bethlehem, because it was the ancestral home, so to speak, because Joseph was of the house and lineage of David.
Think about that for a minute. One of Joseph's ancestors was King David, the great hero of Israel/Judea history (or maybe second to Moses, depending on how you define "hero" I guess). We haven't actually been told anything about Joseph in Luke's account; he hasn't been identified as a carpenter at this point. Still, though, whatever Joseph was at this point, how must it have felt to have only the greatest king your people had ever known in your ancestral line? Did Joseph ever wonder why he wasn't a king himself, or at least some kind of royalty, instead of a guy schlepping across Galilee and into Judea with a pregnant fiancee and some kind of wild story about her pregnancy being "by the Holy Spirit" (really, what does that even mean?)? Joseph, in Luke's account, hasn't had the benefit of any angel visits, in a dream or otherwise. The whole episode must have felt very confusing and confounding to him.
You might imagine Joseph thinking to himself on the whole trip, "Don't have the baby while we're in Bethlehem. Don't have the baby while we're in Bethlehem. Don't have the baby..." ad infinitium while on the road, while seeking a place to stay -- where were these vaunted relatives of the house of David? Was there no one to take him in? Was everybody else skipping out on the registration?
Notice also Luke's last bit of dramatic emphasis; after all, Matthew had nothing to say about a manger, or the lack of room in the inn (by this time Matthew hasn't said anything about the location or circumstances of the birth, and by the time the Magi show up they're in a house of some sort -- Matt. 2:11!), or "swaddling clothes" as the KJV put it. These are all unique to the narrative of the drama master Luke. (It's also worth remembering that in Matthew's narrative Joseph had gone ahead and married Mary, but here that doesn't seem to have happened if one believes verse 5.)
Yes, Luke has more up his sleeve. But freeze the picture here for now. An exhausted mother, a perplexed father, a newborn baby boy resting in a glorified feed trough. And no idea what was going to happen next.